Theorem bj-xpimasn

Description: The image of a singleton, general case. [Change and relabel xpimasn accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-xpimasn	$$⊢\left(A\times B\right)\left[\left\{X\right\}\right]=if\left(X\in A,B,\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		xpima	$$⊢\left(A\times B\right)\left[\left\{X\right\}\right]=if\left(A\cap \left\{X\right\}=\varnothing ,\varnothing ,B\right)$$
2		disjsn	$$⊢A\cap \left\{X\right\}=\varnothing \leftrightarrow \neg X\in A$$
3		eqid	$$⊢B=B$$
4	2 3	ifbieq2i	$$⊢if\left(A\cap \left\{X\right\}=\varnothing ,\varnothing ,B\right)=if\left(\neg X\in A,\varnothing ,B\right)$$
5		ifnot	$$⊢if\left(\neg X\in A,\varnothing ,B\right)=if\left(X\in A,B,\varnothing \right)$$
6	1 4 5	3eqtri	$$⊢\left(A\times B\right)\left[\left\{X\right\}\right]=if\left(X\in A,B,\varnothing \right)$$